Provable computation of L-functions (Dokchitser procedure)

[robertwb]

This is related to #4475.

See also this psage ticket:  http://code.google.com/p/purplesage/issues/detail?id=14

[robertwb]

Depends on #9165, #9184, #9180.

[was]

Why do you write above that this "Depends on #9165"? There is no code there, and that is related to the Cygwin port?

[robertwb]

The dependency should have been #9156 (which is a tiny, now merged, ticket) rather than #9165.

This still needs work in two ways. Firstly, it needs more doctests/documentation (though most of the important/tricky functions are already done), and there is also a bug in computing the tail of the G-function summation that I'm still tracking down. I did some work towards this during Sage Days 22, and even thought about it a touch this week, but haven't quite gotten to the bottom of it.

$ sage -coverage sage/lfunctions/lfunction.py 
----------------------------------------------------------------------
sage/lfunctions/lfunction.py
ERROR: Please add a `TestSuite(s).run()` doctest.
SCORE sage/lfunctions/lfunction.py: 61% (19 of 31)

$ sage -coverage sage/lfunctions/G_function.py 
----------------------------------------------------------------------
sage/lfunctions/G_function.py
ERROR: Please add a `TestSuite(s).run()` doctest.
SCORE sage/lfunctions/G_function.py: 59% (13 of 22)

[robertwb]

This is done except for doctests on some functions (though it could use some optimisations too).

[was]

(I just rebased this with Robert for sage-4.7.2.alpha2.)

[was]

All patches apply (with a minor rebase) and mostly work. However, there is one bug.

Unfortunately, this is wrong, since the coefficients of 1 and T have
to be 0.0000?.  However, the output *is* correct to 10 bits of
precision, so the correct fix is just to truncate.::

    sage: L = LFunction(EllipticCurve('389a'))
    sage: L.taylor_series(RealField(10)(1), 3, proof=True)
    -0.00002125? + 0.00001204?*T + 0.75933?*T^2 - 0.43032?*T^3 + O(T^4)

[was]

(slightly updated)